On the role of meta-accelerations in the evolutional dynamics and stability of weakly-dissipative solids

The presentation deals with the comparative analysis of stability of the torque-free rotation of a rigid body and a solid with internal dissipation. It is shown that the dissipation-induced instability has a critical influence on stability of the rotating body. Lyapunov’s approach to stability is demonstrated to be very sensitive to the mechanical model under consideration. This weak side of Lyapunov’s approach can be overcome by means of introducing the concept of the dissipative rigid body. On some examples we demonstrate as to how the dissipation in a rigid body is introduced and the equations of motion are derived. These equations allow one to study both the stability and the evolutional dynamics of the weakly-damped mechanical systems. The meta-accelerations (the third time derivatives of displacements and rotation angles) are shown to be of a crucial importance for the entire analysis.